An important model system with fascinating theoretical insights to study two-dimensional QSHE is the well-known Kane-Mele model of graphene with spin-orbit couplings (SOCs) [1]}. This model exhibits a pair of helical edge modes with opposite spin-polarization and wave numbers which are connected by time-reversal symmetry.
In the presence of the axial spin symmetry, this model can be considered as two decoupled copies of the Haldane’s model [2]} for electrons with opposite direction of spin. Therefore, in this regime each helical edge mode can be viewed as a chiral edge mode.
If the axial spin symmetry is broken, by adding the Rashba SOC as a common spin-flipping interaction, spin is no longer a good quantum number.
In this regime, each momentum eigenstate can generally be a linear combinations of the spin-up and spin-down eigenstates and hence called generic helical mode [3]}.
